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In 1998 Jean's annual salary was greater than Kevin's annual salary. In each of 1999 and 2000, Jean's annual salary was \(3\) percent greater in that year than in the preceding year and Kevin's annual salary was also \(3\) percent greater in that year than in the preceding year. By what percent did the difference between Jean's and Kevin's annual salaries increase from 1998 to 2000?
Let's start by understanding what we're being asked to find. We have two people, Jean and Kevin, with different salaries in 1998. Both of their salaries grow by exactly 3% each year for two consecutive years (1999 and 2000). The question asks: by what percent did the difference between their salaries increase from 1998 to 2000?
This is asking about the change in the gap between their salaries, not the change in their individual salaries. Since both salaries are growing at the same rate, we need to see how this affects the absolute dollar difference between them.
Process Skill: TRANSLATE - Converting the problem language to focus on the difference between salaries, not the individual salaries themselves
To make this crystal clear, let's use simple numbers. Since Jean's salary was greater than Kevin's in 1998, let's say:
The initial difference between their salaries is: \(\$120,000 - \$100,000 = \$20,000\)
These numbers are chosen to be simple and to clearly show the relationship. The actual dollar amounts don't matter - what matters is that Jean earns more and both salaries grow by the same percentage.
Now let's see what happens when both salaries grow by 3% each year for two years.
After 1999 (one year of 3% growth):
After 2000 (two years of 3% growth):
Notice that the difference itself has grown from \(\$20,000\) to \(\$21,218\).
To find by what percent the difference increased:
Original difference: \(\$20,000\)
Final difference: \(\$21,218\)
Increase in difference: \(\$21,218 - \$20,000 = \$1,218\)
Percentage increase = \(\frac{\text{Increase}}{\text{Original}} \times 100\%\)
Percentage increase = \(\frac{\$1,218}{\$20,000} \times 100\% = 6.09\%\)
Key Insight: When both salaries grow by 3% per year for 2 years, the difference between them grows by the same compound rate: \((1.03)^2 - 1 = 1.0609 - 1 = 0.0609 = 6.09\%\)
This makes intuitive sense: if both quantities grow by the same percentage, their difference grows by that same compounded percentage.
The difference between Jean's and Kevin's annual salaries increased by 6.09% from 1998 to 2000.
This corresponds to answer choice D. 6.09%
1. Misinterpreting what needs to be calculated
Students often focus on calculating the percentage increase in individual salaries (which would simply be the compound growth rate) rather than understanding that the question asks for the percentage increase in the difference between the salaries. This leads them to incorrectly conclude the answer is 6% (simple addition of two 3% increases) or calculate individual salary growth.
2. Assuming the percentage increase in difference equals the sum of annual percentage increases
Many students think that since both salaries increase by 3% each year for 2 years, the difference will increase by 3% + 3% = 6%. They fail to recognize that compound growth applies to the difference as well, missing the compounding effect that leads to the additional 0.09%.
3. Getting confused about whether specific salary values are needed
Some students get stuck thinking they need the actual 1998 salary amounts to solve the problem. They don't realize that the percentage increase in the difference will be the same regardless of the initial salary values, as long as both salaries grow at the same rate.
1. Arithmetic errors in compound growth calculations
When calculating \((1.03)^2 = 1.0609\), students may make computational errors, getting 1.06 instead, which would lead them to select 6% as the answer. Similarly, they might incorrectly calculate \(1.0609 - 1 = 0.609\) instead of \(0.0609\).
2. Incorrectly applying the growth rate
Students may apply 3% growth incorrectly by adding 3% to the previous year's amount instead of multiplying by 1.03, or they may forget to compound the growth properly over the two-year period.
1. Selecting 6% instead of 6.09%
After correctly understanding that they need the compound growth rate, students may calculate \((1.03)^2 - 1\) but then round 6.09% down to 6%, selecting choice C instead of the correct answer D. They miss the importance of the additional 0.09% that comes from compounding.
Step 1: Choose convenient concrete salary values
Let's assign specific dollar amounts that satisfy Jean's salary > Kevin's salary:
• Kevin's 1998 salary: \(\$100,000\)
• Jean's 1998 salary: \(\$130,000\)
These values are chosen because they're easy to work with and clearly satisfy the given constraint.
Step 2: Calculate the initial difference
Initial difference = \(\$130,000 - \$100,000 = \$30,000\)
Step 3: Apply 3% growth for each year
After 1999 (one year of 3% growth):
• Kevin's salary: \(\$100,000 \times 1.03 = \$103,000\)
• Jean's salary: \(\$130,000 \times 1.03 = \$133,900\)
After 2000 (second year of 3% growth):
• Kevin's salary: \(\$103,000 \times 1.03 = \$106,090\)
• Jean's salary: \(\$133,900 \times 1.03 = \$137,917\)
Step 4: Calculate the final difference
Final difference = \(\$137,917 - \$106,090 = \$31,827\)
Step 5: Find the percent increase in the difference
Percent increase = \(\frac{\text{Final difference} - \text{Initial difference}}{\text{Initial difference}} \times 100\%\)
= \(\frac{\$31,827 - \$30,000}{\$30,000} \times 100\%\)
= \(\frac{\$1,827}{\$30,000} \times 100\%\)
= \(0.0609 \times 100\% = 6.09\%\)
Key Insight: When both quantities grow by the same percentage rate, their difference also grows by that same compounded rate. Here, both salaries grew by \((1.03)^2 - 1 = 1.0609 - 1 = 6.09\%\) over two years, so their difference increased by exactly 6.09%.